About Trigonometry

Trigonometry is fascinating!
It started as the measurement (Greek metron) of
triangles (Greek trigonon), but now it has been formalized
under the influence of algebra and analytic geometry and we talk of
trigonometric functions. not just
sides and angles of triangles.

Trig is almost the ideal math subject.
Big and complex enough to have all sorts of interesting odd corners, it
is still small and regular enough to be taught thoroughly in a
semester. (You can easily master the essential points in a week or so.)
It has lots of obvious practical uses, some of which are actually
taught in the usual trig course. And trig extends plenty of tentacles
into other fields like complex numbers, logarithms, and calculus.

If you’d like to learn some of the history of trigonometry and peer
into its dark corners, I recommend Trigonometric Delights
by Eli Maor (Princeton University Press, 1998).

The computations in trigonometry used to be a big obstacle.
But now that we have calculators, that’s no longer an issue.

Would you believe that when I studied trig, back when dinosaurs ruled
the earth (actually, in the 1960s), to solve any problem we had to
look up function values in long tables in the back of the book, and
then multiply or divide those five-place decimals by hand?
The “better” books even included tables of logs of the trig functions,
so that we could save work by adding and subtracting five-place
decimals instead of multiplying and dividing them. My College
Outline Series trig book covered all of plane and spherical
trigonometry in 188 pages—but then needed an additional 138 pages
for the necessary tables!

Though calculators have freed us from tedious computation,
there’s still one big stumbling block in the way many trig courses
are taught: all those identities. They’re just too much to memorize.
(Many students despair of understanding what’s going on, so they just
try to memorize everything and hope for the best at exam time.)
Is it tan² A + sec² A = 1
or tan² A = sec² A + 1?
(Actually, it’s neither—see equation 39!)

Fortunately, you don’t need to memorize them. This paper
shows you the few that you do need to memorize, and how you can
produce the others as needed. I’ll present some ideas of my own, and a
wonderful insight by W.W. Sawyer.

About Trig without Tears

I wrote Trig without Tears to show that
you need to memorize very little. Instead, you learn how all
the pieces of trigonometry hang together, and you get used to
combining identities in different ways so that you can derive most
results on the fly in just a couple of steps.

To help you find things, I’ll number the most important equations
and other facts. (Don’t worry about the gaps in the numbering. I’ve
left those to make it easier to add information to these pages.)

A very few of those, which you need to memorize, will
be marked “memorize”.
Please don’t memorize the others. The whole point of
Trig without Tears
is to teach you how to derive them as needed without
memorizing them. If you can’t think how to derive one, the boxes
should make it easy to find it. But then please work through the
explanation. I truly believe that if you once thoroughly understand
how all these identities hang together, you’ll never have to memorize
them again. (It’s worked for me since I first studied trig in 1965.)

This is rather a long document for reading on screen. If you
prefer, you might want to print the printer-friendly
version.

Abstract or Concrete?

Trig is really a mix of practical and theoretical. On one
hand, you find distances or angles in triangles; that’s pretty
concrete. Computers do it now, but when artillery was guided by humans
they used trig to figure how to aim the cannon. Land surveying still
needs trig, and so do plenty of other fields. If you take a high-school
physics course, you’ll use plenty of trig in it.
Parts 2 through 4 in this book have the practical stuff.

But as you go further in math, though there are usually some
real-world problems there somewhere, you spend a greater and greater
proportion of your time manipulating symbols. That makes sense:
tougher problems take more thought and more calculation to solve.

Traditionally, trig courses are a mix of the practical stuff
with the stuff that has no direct practical application but
that you’ll need to do work in calculus or physics or other
technical fields later. Nearly all the trig identities fall into that
category.
Part 5 and afterward are the groundwork for those future courses.

I don’t say you should stop at the end of Part 4, even
if you’re never planning to take another course. “Is it
useful to me?” isn’t always the right question to ask. (If
people decided things that way, video-game manufacturers would go
out of business tomorrow.)
Better questions are, “Is this interesting? Would I like to
pursue this just for the fun of stretching my mind?”

Yes, even “Is it beautiful?” Symmetry is a
lot of how we judge beauty—if you doubt that, picture someone
good-looking, and then mentally change one side of their face. There
are some beautiful symmetries in the first half of the book, but the
second half has a lot more, plus answers to questions bright students
have always asked, like What’s the
logarithm of a negative number?

Bonus Topics

By the way, I love explaining things but sometimes I go on a bit too
long. So I’ve put some interesting but nonessential notes at the
end of most chapters and inserted hyperlinks to them at appropriate points.
If you follow them (and I hope you will!), use
your browser’s “back” command to return to the
main text.

Much as it pains me to say so, if you’re pressed for time you can
still get all the essential points by ignoring those side notes.
In token of this, they’re labeled BTW.
But you’ll miss some of the fun.

What You Won’t Find Here

Trig without Tears deals exclusively with
plane trigonometry, which is what’s taught today in nearly every first
course. Spherical trigonometry is not part of this book.

I’m also restricting myself to real
arguments to the functions and real values of the functions. I have
to draw the line somewhere! (I do use real-valued functions with
the polar form of complex numbers in the
Notes.)
For complex trigonometric functions, see
chapter 14 of Eli Maor’s Trigonometric delights
(Princeton University Press, 1998).

In Trig without Tears we’ll work with identities and
solve triangles. A separate (and much shorter) page of mine explains
how to solve trigonometric
equations.

Triangles 101

Let’s not take anything for granted. You probably
remember some basic facts from earlier school, but—

An angle is the meeting of two lines at a point. You can
also think of an angle as the amount of rotation a moving line passes
through—a hand on a clock, for instance.

The size of an angle is measured in degrees (or
radians, as you’ll see below). An
angle of 90°, like 3 to 6 on a clock face, is called a
right angle.

In an hour, the minute hand of the clock travels all the way
around the clock, through four right angles, so a full circle is
360°.

A triangle is a plane figure (on a flat surface)
with three straight sides and therefore three interior angles
(angles between two sides, inside the triangle).

The three angles of a triangle always total 180°.

An important theorem, which we won’t prove: Of you consider
two sides and their opposite angles,
the longer side is always opposite the larger angle.

An equilateral triangle is one where all three sides are
equal. Since no side is greater than any other, no angles can be
greater than any other (see previous bullet point), so in an
equilateral triangle each angle is 180°/3 = 60°.

A right triangle is one that contains a right angle
(90°). The long side, which must be opposite the right angle, is
called the hypotenuse, and the short sides are sometimes
called legs.

The famous Pythagorean Theorem or
Theorem of Pythagoras,
a² + b² = c²,
says that if you add the squares of the two legs you get the square of
the hypotenuse.

An acute triangle has only acute angles in it; an
acute angle is one that is greater than 0° and less than
90°.

If a triangle contains an obtuse angle (greater than
90° but less than 180°), it is called, naturally enough, an
obtuse triangle. In a right triangle or obtuse triangle, the
other two triangles must be acute; otherwise the three angles would
total more than 180°.

Notation

Fractions other than ½ are written
using the slash, such as a/b for a over b.

Interval Notation

In talking about the domains and ranges of
functions, it is handy to use interval notation. Thus instead
of saying that x is between 0 and π, we can use the
open interval (0, π) if the endpoints are not included, or the
closed interval [0, π] if the endpoints are included.

You can also have a half-open interval. For instance,
the interval [0, 2π) is all numbers ≥ 0 and
< 2π. You could also say it’s the interval from 0
(inclusive) to 2π (exclusive).

Quadrants

Starting with Part
5, I’ll be referring to the four quadrants of a circle, or
of the xy plane. The division for a circle would be the two
lines from 12 to 6 and from 3 to 9; on the plane, the division is the
x and y axes.

The quadrants are usually given Roman numerals,
starting at the upper right and going counterclockwise. So
Quadrant I or Q I is the top right quarter, Q II is the
top left, Q III is the bottom left, and Q IV is the bottom
right.

Degrees and Radians

These pages will show examples with both radians and degrees. The
same theorems apply to either way of measuring an angle, and you need
to practice with both.

A lot of students seem to find radians terrifying. But measuring
angles in degrees and radians is no worse than measuring temperature
in °F and °C. In fact, angle measure is easier because 0°F
and 0°C are not the same temperature, but 0° and 0 radians are
the same angle.

Just remember that
a complete circle is 2π radians.
Then,
think of the twelve hours numbered around the circumference of a clock face.
When the hour hand goes all the way around, it
travels through 2π radians or 360°.
Six hours is half of that, 2π/2 = π or 180°, one hour is
π/6 or 30°, two hours is
2π/6 = π/3 or 60°, and so on.
(Thanks are due to Jeffrey T. Birt for this suggestion.)

One technical note: Angles don’t actually
have units — they’re dimensionless.
If you say “π radians”, you could just as well say
“π” and leave off the “radians”.

If you’re measuring in degrees, then you do need to use
the degree mark. 180° = π, or π radians. In other
words, that degree mark (°) just means
“×π/180”. It’s the same sort of animal as
the percent sign (%), which really just means “/100”.

So you can convert between degrees and radians exactly the same way
you convert between inches and feet, or between centimeters and
meters. (If conversions in general are a problem for you, you might
like to consult my page on that topic.)

But even though you can convert between degrees and radians, it’s
probably better to learn to think in both. Here’s an analogy.

When you learn a foreign language L, you go through a stage where
you mentally translate what someone says in L into your own language,
formulate your answer in your own language, mentally translate it into
L, and then speak. Eventually you get past that stage, and you carry
on a conversation in the other language without translating. Not only
is it more fun, it’s a heck of a lot faster and easier.

You want to train yourself to work with radians for the same
reason: it’s more efficient, and saving work is always good.
Practice visualizing
an angle of π/6 or 3π/4 or 5π/3 directly,
without translating to degrees. You’ll be surprised how quickly it will become second
nature!

Practice Problems

To get the most benefit from these problems, work them
without first looking at the solutions. Refer back to the chapter
text if you need to refresh your memory.

Recommendation: Work them on paper —
it’s harder to fool yourself about whether you really
understand a problem completely.

2
Which is the correct definition of an acute angle, in interval
notation?
(a) (0°, 90°)
(b) [0°, 90°]

3
Two angles of a triangle are 80° and 40°. Fine the third angle.

4
A triangle has an angle of 90°. The two short sides (next to that
angle) are 5 and 12. Find the third side.

5
Find these angles in radian measure:
(a) 60°
(b) 126°;
(c) 45°.

Where possible, give an exact answer rather than a decimal
approximation.

6
Who said, “The sum of the square roots of any two sides of an
isosceles triangle is equal to the square root of the remaining
side”? Is that correct?

7
On a circular clock face, which numbers are the boundaries of each
quadrant?

BTW: The Problem with Memorizing

Dad sighed.
“Kip, do you think that table was brought down from on
high by an archangel?”

Robert A. Heinlein, in Have Space Suit—Will Travel
(1958)

It’s not just that there are so many trig identities; they seem so
arbitrary. Of course they’re not really arbitrary, since all
can be proved; but when you try to memorize all of them they seem like
a jumble of symbols where the right ones aren’t more obviously right
than the wrong ones. For example, is it
sec² A = 1 + tan² A
or tan² A = 1 + sec² A ?
I doubt you know off hand which is right; I certainly don’t remember.
Who can remember a dozen or more like that, and remember all of them
accurately?

Too many teachers expect (or allow) students to memorize the trig
identities and parrot them on demand, much like a series of
Bible verses. In other words, even if they’re originally taught as a
series of connected propositions, they’re remembered and used as a set
of unrelated facts. And that, I think, is the problem. The trig
identities were not brought down by an archangel; they were
developed by mathematicians, and it’s well within your grasp to
re-develop them when you need to. With effort, we can remember a few
key facts about anything. But it’s much easier if we can fit
them into a context, so that
the identities work together as a whole.

Why bother? Well, of course it will make your life easier in trig
class. But you’ll also need the trig identities in later math classes,
especially calculus, and in physics and engineering classes. In all of
those, you’ll find the going much easier if you’re thoroughly grounded
in trigonometry as a unified field of knowledge instead of a
collection of unrelated facts.

This is why it’s easier to remember almost any song than
an equivalent length of prose: the song gives you additional cues in
the rhythm, common patterns of emphasis, and usually rhymes at
the ends of lines. With prose you have only the general thought to
hold it together, so that you must memorize it essentially as a series
of words. With the song there are internal structures that help you,
even if you’re not aware of them.

If you’re memorizing Lincoln’s Gettysburg Address, you might have
trouble remembering whether he said “recall” or “remember” at a
certain point; in a song, there’s no possible doubt which of those
words is right because the wrong one won’t fit in the rhythm.

On the Other Hand ...

I’m not against all memorization.
Some things have to be memorized because they’re a
matter of definition. Others you may choose to memorize
because you use them very often, you’re confident you can memorize
them correctly, and the derivation takes more time than you’re
comfortable with. Still others you may not set out to memorize, but
after using them many times you find you’ve memorized them without
trying to—much like a telephone number that you dial
often.

I’m not against all memorization; I’m against needless memorization
used as a substitute for thought. If you decide in particular cases
that memory works well for you, I won’t argue. But I do hope you see
the need to be able to re-derive things on the spot, in case your
memory fails. Have you ever dialed a friend’s telephone number and
found you couldn’t quite remember whether it was 6821 or 8621? If
you can’t remember a phone number, you have to look it up in the book.
My goal is to free you from having to look up trig identities in the
book.

Thanks to David Dixon
for an
illuminating exchange of notes on this topic. He made me realize that
I was sounding more anti-memory than I meant to, and in
consequence I’ve added this note. But he may not necessarily agree
with what I say here.

I wrote these pages to show you how to make the trig identities
“fit” as a coherent whole, so that you’ll have no more doubt about
them than you do about the words of a song you know well. The
difference is that you won’t need to do it from memory. And you’ll
gain the sense of power that comes from mastering your subject
instead of groping tentatively and hoping to strike the right answer
by good luck.

Updated the mathematical notation, particularly the use of italics and spaces, to conform to the standard. I used Jukka Korpela’s comprehensive Writing Mathematical Expressions (2014, Suomen E-painos Oy), ISBN 978-952-6613-25-3.